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<title>Atlas software user guide -- Real Weyl groups</title>
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<h2>Real Weyl groups</h2>
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<i>Last updated: October 15, 2005</i>
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Let G be a connected complex reductive algebraic group, and fix a
<a href="realforms.html">real form</a> of G, defined by an involution &theta;.
Let K be the group of &theta;-fixed points in G, and let H be a 
&theta;-invariant Cartan subgroup of G. The <i>real Weyl group</i> associated
to H is the group W(K,T) = N<sub>K</sub>(T)/(K&cap;T), which is also 
W(G(<b>R</b>),H(<b>R</b>)) = N<sub>G(<b>R</b>)</sub>(T)/T(<b>R</b>).
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This group, which is rather delicate to compute, plays an important role in
the representation theory of G(<b>R</b>). Even though it may always be realized
as a subgroup of <b>GL</b>(<b>t</b>) (where <b>t</b> is the Lie algebra of T)
generated by elements of order 2, it is not a Coxeter group in general. It has
the following structure:
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W(K,T)=(W<sub><b>C</b></sub>)<sup>&theta;</sup>.((A.W<sub>ic</sub>) &#215; W<sub><b>R</b></sub>)
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where the symbol &#8220;.&#8221; denotes a semidirect product, the symbol 
&#8220;&#215;&#8221; denotes direct product, and the groups have the following 
meaning:
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W<sub><b>C</b></sub> is the Weyl group of the complex root system
that was mentioned <a href="cartan.html">here</a> (so the 
&#8220;diagonal subgroup&#8221;
(W<sub><b>C</b></sub>)<sup>&theta;</sup>, as a subgroup of
<b>GL</b>(<b>t</b>), is not generated by reflections unless it is trivial);
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<li>
W<sub>ic</sub> is the Weyl group of the system of imaginary compact roots for 
(T,&theta;);
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W<sub><b>R</b></sub> is the Weyl group of the system of real roots for 
(T,&theta;);
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<li>
finally, A is an elementary abelian 2-group, that might be called the 
&#8220;R-group&#8221; for (T,&theta;). To compute it one should look at the 
positive imaginary roots that are orthogonal to the half-sum of positive 
<i>compact</i> imaginary roots; these form a strongly orthogonal set of 
non-compact imaginary roots. For each such root &alpha;, call 
m<sub>&alpha;</sub> the image of -1 under the corresponding coroot; then the 
group A is the set of products of reflections r<sub>&alpha;</sub> for
which the corresponding product of m<sub>&alpha;</sub>'s is 1.
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A subtle aspect is that the m<sub>&alpha;</sub> above, and therefore the group 
A, depend on the choice of covering group. Already in the simple case of type 
A1, for the fundamental Cartan, and the split real form, we have that 
m<sub>&alpha;</sub> is non-trivial (and therefore A is trivial) when G is 
<b>SL</b>(2), whereas m<sub>&alpha;</sub> is trivial (and therefore A is 
non-trivial) when G is <b>PSL</b>(2).
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The &#8220;realweyl&#8221; command will output the real Weyl group for any 
given conjugacy class of Cartan subgroups, together with a set of generators, 
expressed in terms of the standard generators of the Weyl group W.
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<p>
<a href="userintro.html">Back</a> to the introduction.<br>
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